《Linear Algebra Done Wrong》：为高阶学生打造的严谨入门指南

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6 1. Basic Notions

2. Linear combinations, bases.

Let V be a vector space, and let v

, v

, . . . , v

∈ V be a collection of vectors.

A linear combination of vectors v

, v

, . . . , v

is a sum of form

+ α

+ . . . + α

k=1

Deﬁnition 2.1. A system of vectors v

, v

, . . . v

∈ V is called a basis (for

the vector space V ) if any vector v ∈ V admits a unique representation as

a linear combination

v = α

+ α

+ . . . + α

k=1

The coeﬃcients α

, α

, . . . , α

are called coordinates of the vector v (in the

basis, or with respect to the basis v

, v

, . . . , v

Another way to say that v

, v

, . . . , v

is a basis is to say that for any

possible choice of the right side v, the equation x

+. . .+x

= v

(with unknowns x

) has a unique solution.

Before discussing any properties of bases

, let us give few a examples,

showing that such objects exist, and that it makes sense to study them.

Example 2.2. In the ﬁrst example the space V is R

. Consider vectors













, e













, e













, . . . , e













(the vector e

has all entries 0 except the entry number k, which is 1). The

system of vectors e

, e

, . . . , e

is a basis in R

. Indeed, any vector

v =













∈ R

can be represented as the linear combination

v = x

+ x

+ . . . x

k=1

and this representation is unique. The system e

, e

, . . . , e

∈ R

is called

the standard basis in R

the plural for the “basis” is bases, the same as the plural for “base”

2. Linear combinations, bases. 7

Example 2.3. In this example the space is the space P

of the polynomials

of degree at most n. Consider vectors (polynomials) e

, e

, . . . , e

∈ P

deﬁned by

:= 1, e

:= t, e

:= t

, e

:= t

, . . . , e

:= t

Clearly, any polynomial p, p(t) = a

t+a

+. . .+a

admits a unique

representation

p = a

+ a

+ . . . + a

So the system e

, e

, . . . , e

∈ P

is a basis in P

. We will call it the

standard basis in P

Remark 2.4. If a vector space V has a basis v

, v

, . . . , v

, then any vector

v is uniquely deﬁned by its coeﬃcients in the decomposition v =

k=1

. This is a very im-

portant remark, that

will be used through-

out the book. It al-

lows us to translate

any statement about

the standard column

space R

(or, more

generally F

) to a

vector space V with

a basis v

, v

, . . . , v

So, if we stack the coeﬃcients α

in a column, we can operate with them

as if they were column vectors, i.e. as with elements of R

(or F

if V is a

vector space over a ﬁeld F; most important cases are F = R of F = C, but

this also works for general ﬁelds F).

Namely, if v =

k=1

and w =

k=1

, then

v + w =

k=1

(α

+ β

i.e. to get the column of coordinates of the sum one just need to add the

columns of coordinates of the summands. Similarly, to get the coordinates

of αv we need simply to multiply the column of coordinates of v by α.

2.1. Generating and linearly independent systems. The deﬁnition

of a basis says that any vector admits a unique representation as a linear

combination. This statement is in fact two statements, namely that the rep-

resentation exists and that it is unique. Let us analyze these two statements

separately.

If we only consider the existence we get the following notion

Deﬁnition 2.5. A system of vectors v

, v

, . . . , v

∈ V is called a generating

system (also a spanning system, or a complete system) in V if any vector

v ∈ V admits representation as a linear combination

v = α

+ α

+ . . . + α

k=1

The only diﬀerence from the deﬁnition of a basis is that we do not assume

that the representation above is unique.

8 1. Basic Notions

The words generating, spanning and complete here are synonyms. I per-

sonally prefer the term complete, because of my operator theory background.

Generating and spanning are more often used in linear algebra textbooks.

Clearly, any basis is a generating (complete) system. Also, if we have a

basis, say v

, v

, . . . , v

, and we add to it several vectors, say v

n+1

, . . . , v

then the new system will be a generating (complete) system. Indeed, we can

represent any vector as a linear combination of the vectors v

, v

, . . . , v

and just ignore the new ones (by putting corresponding coeﬃcients α

= 0).

Now, let us turn our attention to the uniqueness. We do not want to

worry about existence, so let us consider the zero vector 0, which always

admits a representation as a linear combination.

Deﬁnition. A linear combination α

+ α

+ . . . + α

is called trivial

if α

= 0 ∀k.

A trivial linear combination is always (for all choices of vectors

, v

, . . . , v

) equal to 0, and that is probably the reason for the name.

Deﬁnition. A system of vectors v

, v

, . . . , v

∈ V is called linearly inde-

pendent if only the trivial linear combination (

k=1

with α

= 0 ∀k)

of vectors v

, v

, . . . , v

equals 0.

In other words, the system v

, v

, . . . , v

is linearly independent iﬀ the

equation x

+ x

+ . . . + x

= 0 (with unknowns x

) has only trivial

solution x

= x

= . . . = x

= 0.

If a system is not linearly independent, it is called linearly dependent.

By negating the deﬁnition of linear independence, we get the following

Deﬁnition. A system of vectors v

, v

, . . . , v

is called linearly dependent

if 0 can be represented as a nontrivial linear combination, 0 =

k=1

Non-trivial here means that at least one of the coeﬃcient α

is non-zero.

This can be (and usually is) written as

k=1

|α

| 6= 0.

So, restating the deﬁnition we can say, that a system is linearly depen-

dent if and only if there exist scalars α

, α

, . . . , α

k=1

|α

| 6= 0 such

that

k=1

= 0.

An alternative deﬁnition (in terms of equations) is that a system v

, . . . , v

is linearly dependent iﬀ the equation

+ x

+ . . . + x

= 0

(with unknowns x

) has a non-trivial solution. Non-trivial, once again again

means that at least one of x

is diﬀerent from 0, and it can be written as

k=1

| 6= 0.

10 1. Basic Notions

Proof. We already know that a basis is always linearly independent and

complete, so in one direction the proposition is already proved.

Let us prove the other direction. Suppose a system v

, v

, . . . , v

is lin-

early independent and complete. Take an arbitrary vector v ∈ V . Since the

system v

, v

, . . . , v

is linearly complete (generating), v can be represented

v = α

+ α

+ . . . + α

k=1

We only need to show that this representation is unique.

Suppose v admits another representation

v =

k=1

eα

Then

k=1

(α

− eα

k=1

−

k=1

eα

= v − v = 0.

Since the system is linearly independent, α

− eα

= 0 ∀k, and thus the

representation v = α

+ α

+ . . . + α

is unique. 

Remark. In many textbooks a basis is deﬁned as a complete and linearly

independent system (by Proposition 2.7 this deﬁnition is equivalent to ours).

Although this deﬁnition is more common than one presented in this text, I

prefer the latter. It emphasizes the main property of a basis, namely that

any vector admits a unique representation as a linear combination.

Proposition 2.8. Any (ﬁnite) generating system contains a basis.

Proof. Suppose v

, v

, . . . , v

∈ V is a generating (complete) set. If it is

linearly independent, it is a basis, and we are done.

Suppose it is not linearly independent, i.e. it is linearly dependent. Then

there exists a vector v

which can be represented as a linear combination of

the vectors v

, j 6= k.

Since v

can be represented as a linear combination of vectors v

, j 6= k,

any linear combination of vectors v

, v

, . . . , v

can be represented as a linear

combination of the same vectors without v

(i.e. the vectors v

, 1 ≤ j ≤ p,

j 6= k). So, if we delete the vector v

, the new system will still be a complete

one.

If the new system is linearly independent, we are done. If not, we repeat

the procedure.

Repeating this procedure ﬁnitely many times we arrive to a linearly

independent and complete system, because otherwise we delete all vectors

and end up with an empty set.

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